Solution - Finding the roots of polynomials
Other Ways to Solve
Finding the roots of polynomialsStep by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2". 1 more similar replacement(s).
Step 1 :
Checking for a perfect cube :
1.1 x3-x2-10x-8 is not a perfect cube
Trying to factor by pulling out :
1.2 Factoring: x3-x2-10x-8
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -10x-8
Group 2: x3-x2
Pull out from each group separately :
Group 1: (5x+4) • (-2)
Group 2: (x-1) • (x2)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
1.3 Find roots (zeroes) of : F(x) = x3-x2-10x-8
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is -8.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,8
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 0.00 | x+1 | |||||
| -2 | 1 | -2.00 | 0.00 | x+2 | |||||
| -4 | 1 | -4.00 | -48.00 | ||||||
| -8 | 1 | -8.00 | -504.00 | ||||||
| 1 | 1 | 1.00 | -18.00 | ||||||
| 2 | 1 | 2.00 | -24.00 | ||||||
| 4 | 1 | 4.00 | 0.00 | x-4 | |||||
| 8 | 1 | 8.00 | 360.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x3-x2-10x-8
can be divided by 3 different polynomials,including by x-4
Polynomial Long Division :
1.4 Polynomial Long Division
Dividing : x3-x2-10x-8
("Dividend")
By : x-4 ("Divisor")
| dividend | x3 | - | x2 | - | 10x | - | 8 | ||
| - divisor | * x2 | x3 | - | 4x2 | |||||
| remainder | 3x2 | - | 10x | - | 8 | ||||
| - divisor | * 3x1 | 3x2 | - | 12x | |||||
| remainder | 2x | - | 8 | ||||||
| - divisor | * 2x0 | 2x | - | 8 | |||||
| remainder | 0 |
Quotient : x2+3x+2 Remainder: 0
Trying to factor by splitting the middle term
1.5 Factoring x2+3x+2
The first term is, x2 its coefficient is 1 .
The middle term is, +3x its coefficient is 3 .
The last term, "the constant", is +2
Step-1 : Multiply the coefficient of the first term by the constant 1 • 2 = 2
Step-2 : Find two factors of 2 whose sum equals the coefficient of the middle term, which is 3 .
| -2 | + | -1 | = | -3 | ||
| -1 | + | -2 | = | -3 | ||
| 1 | + | 2 | = | 3 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 2
x2 + 1x + 2x + 2
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+1)
Add up the last 2 terms, pulling out common factors :
2 • (x+1)
Step-5 : Add up the four terms of step 4 :
(x+2) • (x+1)
Which is the desired factorization
Final result :
(x + 2) • (x + 1) • (x - 4)
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